- Email: [email protected]
Simulation of transient transport in amorphous multilayers
Journal of Non-Crystalline Solids 299–302 (2002) 439–443 www.elsevier.com/locate/jnoncrysol
Simulation of transient transport in amorphous multilayers A. Picos-Vega a, R. Ramırez-Bon
b,*
a
Centro de Ciencias de la Materia Condensada, Universidad Nacional Aut onoma de M exico, Apdo. Postal 2681, 22800 Ensenada, BC, Mexico b Centro de Investigaci on y de Estudios Avanzados del I.P.N. Unidad Quer etaro, Apdo. Postal 1-1010, 76001 Quer etaro, Qro., Mexico
Abstract We simulate the charge carrier transient transport in amorphous layers of multilayer structures. We studied the effect of the time delay of carriers at interfaces on the transient photocurrent shape. The characteristics of transient photocurrents on amorphous multilayers, as obtained by the time-of-flight (TOF) technique, were simulated by random walks through cubic lattices. We compared our results with experimental data measured on Se=Se1x Tex multilayers. We suggest the existence of two conduction mechanisms involved in the carrier transport through the multilayers. One mechanism is hopping through localized states of the amorphous layers and the other could be thermal activation at interfaces or direct tunnelling through barriers. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 72.10.)d; 72.20. Jv; 73.40.)c
1. Introduction The electrical transient transport in disordered solid materials such as amorphous semiconductors, polymers, chalcogenide glasses, etc. is nonGaussian or dispersive [1–3]. The characteristics of dispersive transport are revealed by the time-offlight (TOF) experiments performed on these types of materials. In this experiment a light pulse absorbed by the material in the adjacent region to a biased electrode generates electron–hole pairs. Depending on the polarity of the bias voltage between electrodes, the holes (electrons) are immediately absorbed at the illuminated electrode, leaving a sheet of electrons (holes) moving across the sample to the collecting electrode. The shape of *
Corresponding author. E-mail address: [email protected] (R. Ramırez-Bon).
the induced photocurrent provides important information about the transport properties of the material such as drift mobility, density of trapping centres, etc. In particular, the photocurrents measured in disordered solids show that as the charge packet moves across the sample, it broadens to a much greater extent than would be expected from diffusion alone. This anomalous behaviour in the dispersion of the travelling charge packet has been explained by several models, namely, the hopping model [2], trap-controlled hopping model [4] and multiple trapping model [5]. All these transport models predict well the hyperbolic shape of the empirically observed photocurrents in disordered materials, which for a single layer can be described by the expression [2] tð1a1 Þ ; t > ts ; iðtÞ ð1Þ tð1a2 Þ ; t < ts ;
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 1 1 7 4 - 7
440
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
where a1 and a2 are the dispersion parameters at short times and at long times, respectively. The time needed for the carriers to cross the sample is known as transit time ts . According to the hopping model, charge carriers move by tunnelling between localized band tail states. The origin of the charge packet dispersion is due to the random distribution of the hopping distances, which produces a wide waiting time distribution of carriers in the localized states. The trap-controlled hopping model considers, additionally, the existence of deep trapping centres randomly distributed in space. In the multiple trapping model the charge carriers move by continually trapping from conduction to localized states and releasing from localized to conduction states. The charge packet dispersion is caused by the energy distribution of localized states. In the last few years, the deposition of amorphous multilayer structures has been done in order to improve some of the transport properties and to build devices with specific characteristics [6,7]. The transient transport in these multilayer structures has also been studied by the TOF technique, showing the dispersive transport characteristics. However, the dispersive transport models formulated for single layers have no direct generalization to these systems. Furthermore, due to the complexity of these models, their generalization to system composed of more than two layers seems rather difficult. In this paper we use Monte Carlo simulation to obtain transient photocurrents in amorphous multilayer structures. The basis of the calculation is the simulation of transient currents induced by the random walk of charge carriers on percolation clusters [8,9]. The modifications of this single layer model to include multilayer systems are explained below. The simulation parameters were adjusted to reproduce some experimental results of transient photocurrents for Se=Se1x Tex [10] multilayers.
2. Simulation The basis of the model is that the energy fluctuations of the band edge in amorphous materials or the distribution of energy barrier heights between hopping sites can be replaced by the topo-
logical disorder of the available paths for the charge transport through the material [8,9]. The complexity of the topological disorder produces the charge packet dispersion. The simulation is based on the following picture: the photogenerated charge packet is represented by a set of particles moving in a random walk through a disordered arrangement of sites, which represents a disordered distribution of localized states available for the carriers’ hopping. The random arrangement of sites is called the percolation cluster [11] and it is constructed by choosing at random a fraction p of sites in a cubic lattice. Because the connectivity of states for the transport is necessary, the fraction p of sites must be always greater than the corresponding percolation threshold, pc , for a cubic lattice connecting sites to first nearest neighbours [11]. The set of particles is introduced into the percolation cluster in a face of the cube, the particles move in the x direction by the action of a bias voltage and are collected at the opposite face. The effect of the bias voltage is considered in the jump probability of carriers to nearest neighbours in the x direction. A complete description of the steps necessary to simulate transient currents in single layers with this model can be found in [8,9,12]. To obtain the transient photocurrents in amorphous multilayer structures we simulated the photocurrents for a structure composed of 10 cubic layers of width L with a fraction of available sites for the transport, p, of about 0.52. The charge carriers move in each layer according to their corresponding physical properties. However, in the calculation we assumed that the connectivity of localized states remains the same for all the states of the multilayer structure at the same temperature, but the drift velocity and the mobility edge change going from one layer to other. This way, the total simulated photocurrent is the sum of all the individual photocurrents induced in each layer of the complete structure [7,13]. The difference in the mobility edges between consecutive layers results in the formation of multiple well (barriers) potential in the sample. The effects of the waiting time of charge carriers to pass over the barriers were introduced in the algorithm by assuming thermal activation with time
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
distribution of probability of the form [12,14] et=s , with s being the characteristic time at which the thermal activation occurs. This exponential activation rate was also used to calculate the release time of carriers at the contact–layer interface. In the algorithm the time delay probability distribution function P ðtÞ is used to evaluate the time a carrier has to wait at an interface before it crosses from one layer to other, or from the contact to the first layer. The characteristic time s was different for carriers released from the contact–layer interface than that for carriers passing over layer–layer interfaces. Once in the first layer, the carriers move through the amorphous layer to the next layer. The process is repeated until the carriers arrive to the end of the sample where they are absorbed by the contact.
3. Results First, we simulated the transient photocurrent in a single layer without charge trapping at the contact–layer interface. The result is shown in Fig. 1
Fig. 1. Transient photocurrent and number of particles inside the percolation cluster (single layer) when no charge is trapped at the contact–layer interface.
441
where are plotted the photocurrent IðnÞ and the number of particles moving inside the percolation cluster N ðnÞ, as a function of the step number n. As can be seen, the photocurrent shape fits very well Eq. (1) and the main features of dispersive transport are reproduced by this simulation. In Fig. 2 the photocurrent and the number of particles in the cluster are shown, for the case when charge carriers are trapped at the contact–layer interface after being photogenerated. In this case, the number of particles at t ¼ 0 ðn ¼ 0Þ inside the percolation cluster is zero because at this time they are all trapped. As time ðnÞ increases, N increases because the probability of the carriers to enter in the percolation cluster increases with time. When all the particles are in the percolation cluster the induced photocurrent is maximum and after the first particles arrive to the absorbing electrode, the photocurrent decreases. As a consequence, the photocurrent for this case is peak-shaped. In Fig. 3 it is shown in log–log scales the simulated photocurrent for the 10-layer structure without the delay time of charge at the contact–layer
Fig. 2. Transient photocurrent and number of particles inside the percolation cluster (single layer) when charge is trapped at the contact–layer interface.
442
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
Fig. 3. Simulated transient photocurrent (solid line) in a 10layer structure without charge trapping at interfaces. The curves labelled with numbers from 1 to 10 correspond to the photocurrent in each layer. The scatter graph is the experimental photocurrent from [10].
interface or the layer–layer interface. It was considered to be the same mobility for all the layers, which is equivalent to having one layer 10 times thicker than a single one of the multilayer structure. Plotted are the individual photocurrents induced in each one of the 10 layers, enumerated from 1 to 10. The total photocurrent induced in the multilayer is the sum of the photocurrents induced in each single layer. We can observe that the induced photocurrent in the first layer behaves as predicted by Eq. (1) and is shown in Fig. 1. None of the remaining nine photocurrents have this behaviour, instead they all have the shape of peak similar to that shown in Fig. 2. This can be explained by the delay time of the carriers in the previous k 1 layer before they arrive to the k layer and contribute to the induced photocurrent in this layer. For comparison, there is also shown, plotted in empty circles, the experimental photocurrent measured in Se=Se1x Tex multilayers [10]. Both simulated and experimental curves are normalized at the transit time ts . There is a disagreement between the simulated and experimental results at short and long times. The next step in the simulation was to consider the time delay of charge carriers at the contact– sample interface. The simulated photocurrents for this case are shown in Fig. 4. As expected the
Fig. 4. Simulated transient photocurrent (solid line) in a 10layer structure considering charge trapping at the contact– sample interface.
simulated photocurrent starts at zero at t ¼ 0, with a shape of peak. The induced photocurrents in each layer, including the first one, are also peakshaped and they are labelled from 1 to 10. The amplitude of these photocurrents is approximately in the same order because the charge flows without delay at the interfaces of the multilayer structure. Comparing the simulated photocurrent with the experimental one, a disagreement at short times is still observed. The last step was to consider a mismatch mobility edge at each interface by including in the simulation a time delay function at position where the mobility edge changes from a lower to a higher value. The time delay distribution function at the contact–sample interface was kept in the calculation. In Fig. 5 are shown the results of this simulation. It is observed that the simulated photocurrent fits very well the double peak shape of the experimental data. By the shape of the induced photocurrents in each one of the layers it is clear that the first peak of the total photocurrent is due mainly to the photocurrents induced in the first three layers. This photocurrent peak is controlled by the time delay of carriers at the contact–sample interface. The sum of the small photocurrent contributions from the rest of the layers originates the second peak of the total photocurrent. From these results we can conclude that the double peak shape of the experimental
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
443
mechanisms involved in the carrier transport through the multilayer structure. The first one is hopping between localized states of the disordered layers and the second could be thermal activation at interfaces or direct tunnelling through barriers. Our simulation results are compatible with a thermal activation mechanism of carriers over a well-defined energy barrier, whose height defines the characteristic waiting time at interfaces.
Acknowledgements
Fig. 5. Simulated transient photocurrent (solid line) in a 10layer structure considering charge trapping at the contact– sample and layer–layer interfaces.
This work was partially supported by CONACyT-Mexico (Project No. 34514-U).
References photocurrent arises from the delay of time of carriers at sample–contact and layer–layer interfaces.
4. Conclusions We reported the transient photocurrents in disordered multilayer systems simulated by the Monte Carlo method. By comparing the simulated results with experimental photocurrents measured in Se=Se1x Tex multilayer system, we found that the photocurrents’ shape is controlled by the time delay of carriers at the interfaces. The time that carriers spend at interfaces contributes to slow down their movement but it is the disordered distribution of sites which contributes to the decaying current slope. Thus, we identified two conduction
[1] H. Scher, M.F. Shlesinger, J.T. Bendler, Phys. Today 26 (1991). [2] H. Scher, H. Montroll, Phys. Rev. B 12 (1975) 2455. [3] G. Pfister, Phys. Rev. Lett. 33 (1974) 1474. [4] G. Pfister, S. Grammatica, J. Mort, Phys. Rev. Lett. 37 (1976) 1360. [5] J. Noolandi, Phys. Rev. B 16 (1972) 4466. [6] L.B. Lin, S.A. Jenekhe, R.H. Young, P.M. Borsenberger, Appl. Phys. Lett. 70 (1997) 2052. [7] V.I. Mikla, Phys. Stat. Sol. (a) 165 (1998) 427. [8] K. Murayama, Philos. Mag. 65 (1992) 749. [9] K. Murayama, M. Mori, Philos. Mag. 65 (1992) 501. [10] S. Imamura, Y. Kanemitsu, M. Saito, H. Sugimoto, J. Non-Cryst. Solids 114 (1989) 121. [11] D. Stauffer, A. Aharony, Introducion to Percolation Theory, Taylor and Francis, London, 1992. [12] A. Picos-Vega, O. Zelaya-Angel, R. Ramırez-Bon, F.J. Espinoza-Beltran, Phys. Rev. B 58 (1998) 14845. [13] L.B. Lin, R.H. Young, M.G. Mason, S.A. Jenekhe, P.M. Borsenberger, Appl. Phys. Lett. 72 (1998) 864. [14] M. Abkowitz, H. Scher, Philos. Mag. 35 (1977) 1585.
Simulation of transient transport in amorphous multilayers A. Picos-Vega a, R. Ramırez-Bon
b,*
a
Centro de Ciencias de la Materia Condensada, Universidad Nacional Aut onoma de M exico, Apdo. Postal 2681, 22800 Ensenada, BC, Mexico b Centro de Investigaci on y de Estudios Avanzados del I.P.N. Unidad Quer etaro, Apdo. Postal 1-1010, 76001 Quer etaro, Qro., Mexico
Abstract We simulate the charge carrier transient transport in amorphous layers of multilayer structures. We studied the effect of the time delay of carriers at interfaces on the transient photocurrent shape. The characteristics of transient photocurrents on amorphous multilayers, as obtained by the time-of-flight (TOF) technique, were simulated by random walks through cubic lattices. We compared our results with experimental data measured on Se=Se1x Tex multilayers. We suggest the existence of two conduction mechanisms involved in the carrier transport through the multilayers. One mechanism is hopping through localized states of the amorphous layers and the other could be thermal activation at interfaces or direct tunnelling through barriers. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 72.10.)d; 72.20. Jv; 73.40.)c
1. Introduction The electrical transient transport in disordered solid materials such as amorphous semiconductors, polymers, chalcogenide glasses, etc. is nonGaussian or dispersive [1–3]. The characteristics of dispersive transport are revealed by the time-offlight (TOF) experiments performed on these types of materials. In this experiment a light pulse absorbed by the material in the adjacent region to a biased electrode generates electron–hole pairs. Depending on the polarity of the bias voltage between electrodes, the holes (electrons) are immediately absorbed at the illuminated electrode, leaving a sheet of electrons (holes) moving across the sample to the collecting electrode. The shape of *
Corresponding author. E-mail address: [email protected] (R. Ramırez-Bon).
the induced photocurrent provides important information about the transport properties of the material such as drift mobility, density of trapping centres, etc. In particular, the photocurrents measured in disordered solids show that as the charge packet moves across the sample, it broadens to a much greater extent than would be expected from diffusion alone. This anomalous behaviour in the dispersion of the travelling charge packet has been explained by several models, namely, the hopping model [2], trap-controlled hopping model [4] and multiple trapping model [5]. All these transport models predict well the hyperbolic shape of the empirically observed photocurrents in disordered materials, which for a single layer can be described by the expression [2] tð1a1 Þ ; t > ts ; iðtÞ ð1Þ tð1a2 Þ ; t < ts ;
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 1 1 7 4 - 7
440
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
where a1 and a2 are the dispersion parameters at short times and at long times, respectively. The time needed for the carriers to cross the sample is known as transit time ts . According to the hopping model, charge carriers move by tunnelling between localized band tail states. The origin of the charge packet dispersion is due to the random distribution of the hopping distances, which produces a wide waiting time distribution of carriers in the localized states. The trap-controlled hopping model considers, additionally, the existence of deep trapping centres randomly distributed in space. In the multiple trapping model the charge carriers move by continually trapping from conduction to localized states and releasing from localized to conduction states. The charge packet dispersion is caused by the energy distribution of localized states. In the last few years, the deposition of amorphous multilayer structures has been done in order to improve some of the transport properties and to build devices with specific characteristics [6,7]. The transient transport in these multilayer structures has also been studied by the TOF technique, showing the dispersive transport characteristics. However, the dispersive transport models formulated for single layers have no direct generalization to these systems. Furthermore, due to the complexity of these models, their generalization to system composed of more than two layers seems rather difficult. In this paper we use Monte Carlo simulation to obtain transient photocurrents in amorphous multilayer structures. The basis of the calculation is the simulation of transient currents induced by the random walk of charge carriers on percolation clusters [8,9]. The modifications of this single layer model to include multilayer systems are explained below. The simulation parameters were adjusted to reproduce some experimental results of transient photocurrents for Se=Se1x Tex [10] multilayers.
2. Simulation The basis of the model is that the energy fluctuations of the band edge in amorphous materials or the distribution of energy barrier heights between hopping sites can be replaced by the topo-
logical disorder of the available paths for the charge transport through the material [8,9]. The complexity of the topological disorder produces the charge packet dispersion. The simulation is based on the following picture: the photogenerated charge packet is represented by a set of particles moving in a random walk through a disordered arrangement of sites, which represents a disordered distribution of localized states available for the carriers’ hopping. The random arrangement of sites is called the percolation cluster [11] and it is constructed by choosing at random a fraction p of sites in a cubic lattice. Because the connectivity of states for the transport is necessary, the fraction p of sites must be always greater than the corresponding percolation threshold, pc , for a cubic lattice connecting sites to first nearest neighbours [11]. The set of particles is introduced into the percolation cluster in a face of the cube, the particles move in the x direction by the action of a bias voltage and are collected at the opposite face. The effect of the bias voltage is considered in the jump probability of carriers to nearest neighbours in the x direction. A complete description of the steps necessary to simulate transient currents in single layers with this model can be found in [8,9,12]. To obtain the transient photocurrents in amorphous multilayer structures we simulated the photocurrents for a structure composed of 10 cubic layers of width L with a fraction of available sites for the transport, p, of about 0.52. The charge carriers move in each layer according to their corresponding physical properties. However, in the calculation we assumed that the connectivity of localized states remains the same for all the states of the multilayer structure at the same temperature, but the drift velocity and the mobility edge change going from one layer to other. This way, the total simulated photocurrent is the sum of all the individual photocurrents induced in each layer of the complete structure [7,13]. The difference in the mobility edges between consecutive layers results in the formation of multiple well (barriers) potential in the sample. The effects of the waiting time of charge carriers to pass over the barriers were introduced in the algorithm by assuming thermal activation with time
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
distribution of probability of the form [12,14] et=s , with s being the characteristic time at which the thermal activation occurs. This exponential activation rate was also used to calculate the release time of carriers at the contact–layer interface. In the algorithm the time delay probability distribution function P ðtÞ is used to evaluate the time a carrier has to wait at an interface before it crosses from one layer to other, or from the contact to the first layer. The characteristic time s was different for carriers released from the contact–layer interface than that for carriers passing over layer–layer interfaces. Once in the first layer, the carriers move through the amorphous layer to the next layer. The process is repeated until the carriers arrive to the end of the sample where they are absorbed by the contact.
3. Results First, we simulated the transient photocurrent in a single layer without charge trapping at the contact–layer interface. The result is shown in Fig. 1
Fig. 1. Transient photocurrent and number of particles inside the percolation cluster (single layer) when no charge is trapped at the contact–layer interface.
441
where are plotted the photocurrent IðnÞ and the number of particles moving inside the percolation cluster N ðnÞ, as a function of the step number n. As can be seen, the photocurrent shape fits very well Eq. (1) and the main features of dispersive transport are reproduced by this simulation. In Fig. 2 the photocurrent and the number of particles in the cluster are shown, for the case when charge carriers are trapped at the contact–layer interface after being photogenerated. In this case, the number of particles at t ¼ 0 ðn ¼ 0Þ inside the percolation cluster is zero because at this time they are all trapped. As time ðnÞ increases, N increases because the probability of the carriers to enter in the percolation cluster increases with time. When all the particles are in the percolation cluster the induced photocurrent is maximum and after the first particles arrive to the absorbing electrode, the photocurrent decreases. As a consequence, the photocurrent for this case is peak-shaped. In Fig. 3 it is shown in log–log scales the simulated photocurrent for the 10-layer structure without the delay time of charge at the contact–layer
Fig. 2. Transient photocurrent and number of particles inside the percolation cluster (single layer) when charge is trapped at the contact–layer interface.
442
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
Fig. 3. Simulated transient photocurrent (solid line) in a 10layer structure without charge trapping at interfaces. The curves labelled with numbers from 1 to 10 correspond to the photocurrent in each layer. The scatter graph is the experimental photocurrent from [10].
interface or the layer–layer interface. It was considered to be the same mobility for all the layers, which is equivalent to having one layer 10 times thicker than a single one of the multilayer structure. Plotted are the individual photocurrents induced in each one of the 10 layers, enumerated from 1 to 10. The total photocurrent induced in the multilayer is the sum of the photocurrents induced in each single layer. We can observe that the induced photocurrent in the first layer behaves as predicted by Eq. (1) and is shown in Fig. 1. None of the remaining nine photocurrents have this behaviour, instead they all have the shape of peak similar to that shown in Fig. 2. This can be explained by the delay time of the carriers in the previous k 1 layer before they arrive to the k layer and contribute to the induced photocurrent in this layer. For comparison, there is also shown, plotted in empty circles, the experimental photocurrent measured in Se=Se1x Tex multilayers [10]. Both simulated and experimental curves are normalized at the transit time ts . There is a disagreement between the simulated and experimental results at short and long times. The next step in the simulation was to consider the time delay of charge carriers at the contact– sample interface. The simulated photocurrents for this case are shown in Fig. 4. As expected the
Fig. 4. Simulated transient photocurrent (solid line) in a 10layer structure considering charge trapping at the contact– sample interface.
simulated photocurrent starts at zero at t ¼ 0, with a shape of peak. The induced photocurrents in each layer, including the first one, are also peakshaped and they are labelled from 1 to 10. The amplitude of these photocurrents is approximately in the same order because the charge flows without delay at the interfaces of the multilayer structure. Comparing the simulated photocurrent with the experimental one, a disagreement at short times is still observed. The last step was to consider a mismatch mobility edge at each interface by including in the simulation a time delay function at position where the mobility edge changes from a lower to a higher value. The time delay distribution function at the contact–sample interface was kept in the calculation. In Fig. 5 are shown the results of this simulation. It is observed that the simulated photocurrent fits very well the double peak shape of the experimental data. By the shape of the induced photocurrents in each one of the layers it is clear that the first peak of the total photocurrent is due mainly to the photocurrents induced in the first three layers. This photocurrent peak is controlled by the time delay of carriers at the contact–sample interface. The sum of the small photocurrent contributions from the rest of the layers originates the second peak of the total photocurrent. From these results we can conclude that the double peak shape of the experimental
A. Picos-Vega, R. Ramırez-Bon / Journal of Non-Crystalline Solids 299–302 (2002) 439–443
443
mechanisms involved in the carrier transport through the multilayer structure. The first one is hopping between localized states of the disordered layers and the second could be thermal activation at interfaces or direct tunnelling through barriers. Our simulation results are compatible with a thermal activation mechanism of carriers over a well-defined energy barrier, whose height defines the characteristic waiting time at interfaces.
Acknowledgements
Fig. 5. Simulated transient photocurrent (solid line) in a 10layer structure considering charge trapping at the contact– sample and layer–layer interfaces.
This work was partially supported by CONACyT-Mexico (Project No. 34514-U).
References photocurrent arises from the delay of time of carriers at sample–contact and layer–layer interfaces.
4. Conclusions We reported the transient photocurrents in disordered multilayer systems simulated by the Monte Carlo method. By comparing the simulated results with experimental photocurrents measured in Se=Se1x Tex multilayer system, we found that the photocurrents’ shape is controlled by the time delay of carriers at the interfaces. The time that carriers spend at interfaces contributes to slow down their movement but it is the disordered distribution of sites which contributes to the decaying current slope. Thus, we identified two conduction
[1] H. Scher, M.F. Shlesinger, J.T. Bendler, Phys. Today 26 (1991). [2] H. Scher, H. Montroll, Phys. Rev. B 12 (1975) 2455. [3] G. Pfister, Phys. Rev. Lett. 33 (1974) 1474. [4] G. Pfister, S. Grammatica, J. Mort, Phys. Rev. Lett. 37 (1976) 1360. [5] J. Noolandi, Phys. Rev. B 16 (1972) 4466. [6] L.B. Lin, S.A. Jenekhe, R.H. Young, P.M. Borsenberger, Appl. Phys. Lett. 70 (1997) 2052. [7] V.I. Mikla, Phys. Stat. Sol. (a) 165 (1998) 427. [8] K. Murayama, Philos. Mag. 65 (1992) 749. [9] K. Murayama, M. Mori, Philos. Mag. 65 (1992) 501. [10] S. Imamura, Y. Kanemitsu, M. Saito, H. Sugimoto, J. Non-Cryst. Solids 114 (1989) 121. [11] D. Stauffer, A. Aharony, Introducion to Percolation Theory, Taylor and Francis, London, 1992. [12] A. Picos-Vega, O. Zelaya-Angel, R. Ramırez-Bon, F.J. Espinoza-Beltran, Phys. Rev. B 58 (1998) 14845. [13] L.B. Lin, R.H. Young, M.G. Mason, S.A. Jenekhe, P.M. Borsenberger, Appl. Phys. Lett. 72 (1998) 864. [14] M. Abkowitz, H. Scher, Philos. Mag. 35 (1977) 1585.